Something first, be careful with the formats of every webpage. These are the most common:
AO3 (At archiveofourown.org): There are chapters like usual, but also "series". Some stories are series with lots of individual fanfic entries, and some items of the series have more than one chapter in themselves. I will link the whole series here in that case but
just click the "Entire Work" button at the top of the page when you open something to make things simpler.
HS Kinkmeme (at homesmut.livejournal.com): Meme fills are in comments with character limits so they are usually divided into lots of tiny parts. Actual fanction content is usually a reply with a title and a "1/??" number, but without the "Re:" at the beginning. If it has the "re:" it's just a comment, don't worry about those, just click the title of the next numbered part you need.
The rest are either regular lj entries or tumblr posts so I don't think there is a lot to say there.
BEWARE, the "SAFE" tag is only related to sexual themes. Stories with violence, blood, etc may still be called safe. Read the warnings on each fic if you have trouble with some topics.
OK, NOW, FANFIC RECS organized by sort-of importance/awesome and type. Fan-adventure styled works and comics are here too even if they are not fanfiction per-se.
( Say goodbye to your free time )